Arithmetic Sequences Notes and Worksheets - Lindsay Bowden - Free Printable

Problem Analysis and Solution

The provided worksheet focuses on arithmetic sequences. Let's solve each part step by step.

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#### Definitions and Notes
1. Sequence: A list of numbers in a particular order.
- Examples: \(\{2, 4, 6, 8, \ldots\}\), \(\{1, 3, 9, 27\}\).

2. Infinite Sequence: A sequence that goes on forever.
- Example: \(\{1, 3, 5, 7, 9, \ldots\}\).

3. Finite Sequence: A sequence that does not go on forever.
- Example: \(\{2, 4, 8, 16\}\).

4. Arithmetic Sequence: A sequence created by adding or subtracting the same number to get to the next term.
- Example: \(\{6, 9, 12, 15, \ldots\}\).

5. Common Difference (d): The number that is added or subtracted to get to the next term in the sequence.
- Example: \(\{5, 10, 15, 20, \ldots\}\) (common difference \(d = 5\)).
- Example: \(\{9, 7, 5, 3, \ldots\}\) (common difference \(d = -2\)).

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#### Examples

##### 1. Is the sequence finite or infinite?
Sequence: \(\{9, 12, 15, 18\}\)

- The sequence is explicitly listed with a finite number of terms (\(9, 12, 15, 18\)).
- There is no indication that it continues indefinitely.

Answer: Finite.

##### 2. Find the next three terms in the sequence.
Sequence: \(\{8, 2, -4, -10, \ldots\}\)

- First, find the common difference (\(d\)):
\[
d = 2 - 8 = -6
\]
Verify:
\[
-4 - 2 = -6, \quad -10 - (-4) = -6
\]
The common difference is \(d = -6\).

- Next terms:
\[
\text{Next term} = -10 + (-6) = -16
\]
\[
\text{Next term} = -16 + (-6) = -22
\]
\[
\text{Next term} = -22 + (-6) = -28
\]

Answer: \(-16, -22, -28\).

##### 3. What is the common difference in this sequence?
Sequence: \(\{54, 63, 72, 81, \ldots\}\)

- Find the common difference (\(d\)):
\[
d = 63 - 54 = 9
\]
Verify:
\[
72 - 63 = 9, \quad 81 - 72 = 9
\]

Answer: \(9\).

##### 4. Is this sequence an arithmetic sequence?
Sequence: \(\{108, 90, 72, 54, 36, \ldots\}\)

- Find the differences between consecutive terms:
\[
90 - 108 = -18
\]
\[
72 - 90 = -18
\]
\[
54 - 72 = -18
\]
\[
36 - 54 = -18
\]

- The differences are constant (\(d = -18\)).

Answer: Yes, it is an arithmetic sequence.

##### 5. What are the next 3 terms in the sequence?
Sequence: \(\{3.5, 4.9, 6.3, 7.7, \ldots\}\)

- Find the common difference (\(d\)):
\[
d = 4.9 - 3.5 = 1.4
\]
Verify:
\[
6.3 - 4.9 = 1.4, \quad 7.7 - 6.3 = 1.4
\]

- Next terms:
\[
\text{Next term} = 7.7 + 1.4 = 9.1
\]
\[
\text{Next term} = 9.1 + 1.4 = 10.5
\]
\[
\text{Next term} = 10.5 + 1.4 = 11.9
\]

Answer: \(9.1, 10.5, 11.9\).

##### 6. What is the common difference in this arithmetic sequence?
Sequence: \(\left\{\frac{1}{3}, \frac{2}{3}, 1, \frac{4}{3}, \ldots\right\}\)

- Convert all terms to have a common denominator (3):
\[
\left\{\frac{1}{3}, \frac{2}{3}, \frac{3}{3}, \frac{4}{3}, \ldots\right\}
\]

- Find the common difference (\(d\)):
\[
d = \frac{2}{3} - \frac{1}{3} = \frac{1}{3}
\]
Verify:
\[
\frac{3}{3} - \frac{2}{3} = \frac{1}{3}, \quad \frac{4}{3} - \frac{3}{3} = \frac{1}{3}
\]

Answer: \(\frac{1}{3}\).

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Final Answers

1. Finite
2. \(-16, -22, -28\)
3. \(9\)
4. Yes
5. \(9.1, 10.5, 11.9\)
6. \(\frac{1}{3}\)

\[
\boxed{
\text{1. Finite, 2. } -16, -22, -28, \text{ 3. } 9, \text{ 4. Yes, 5. } 9.1, 10.5, 11.9, \text{ 6. } \frac{1}{3}
}
\]